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Properties

Label 26T27
26T27 1 6 1->6 16 1->16 2 9 2->9 25 2->25 3 12 3->12 21 3->21 4 4->2 17 4->17 5 26 5->26 8 6->8 22 6->22 7 11 7->11 18 7->18 8->1 14 8->14 9->4 23 9->23 10 10->7 19 10->19 11->10 15 11->15 13 12->13 24 12->24 13->3 20 13->20 14->11 14->16 15->7 15->17 16->3 16->18 17->12 17->19 18->8 18->20 19->4 19->21 20->22 21->9 21->23 22->5 22->24 23->1 23->25 24->10 24->26 25->6 25->14 26->2 26->15
Degree $26$
Order $3042$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_{13}^2:(C_3\times S_3)$

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Copy content magma:G := TransitiveGroup(26, 27);
 

Group invariants

Abstract group:  $C_{13}^2:(C_3\times S_3)$
Copy content magma:IdentifyGroup(G);
 
Order:  $3042=2 \cdot 3^{2} \cdot 13^{2}$
Copy content magma:Order(G);
 
Cyclic:  no
Copy content magma:IsCyclic(G);
 
Abelian:  no
Copy content magma:IsAbelian(G);
 
Solvable:  yes
Copy content magma:IsSolvable(G);
 
Nilpotency class:   not nilpotent
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $26$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $27$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $-1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $1$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,16,3,21,9,23)(2,25,6,22,5,26)(4,17,12,24,10,19)(7,18,8,14,11,15)(13,20)$, $(1,6,8)(2,9,4)(3,12,13)(7,11,10)(14,16,18,20,22,24,26,15,17,19,21,23,25)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $S_3$, $C_6$
$18$:  $S_3\times C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 13: None

Low degree siblings

39T39, 39T40

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{26}$ $1$ $1$ $0$ $()$
2A $2^{13}$ $39$ $2$ $13$ $( 1,24)( 2,14)( 3,17)( 4,20)( 5,23)( 6,26)( 7,16)( 8,19)( 9,22)(10,25)(11,15)(12,18)(13,21)$
3A1 $3^{4},1^{14}$ $26$ $3$ $8$ $( 1, 4, 5)( 2,13, 8)( 3, 9,11)( 6,10, 7)$
3A-1 $3^{4},1^{14}$ $26$ $3$ $8$ $( 1, 5, 4)( 2, 8,13)( 3,11, 9)( 6, 7,10)$
3B1 $3^{8},1^{2}$ $169$ $3$ $16$ $( 1, 2, 5)( 3, 8,10)( 4,11, 6)( 9,13,12)(14,23,24)(15,26,20)(17,19,25)(18,22,21)$
3B-1 $3^{8},1^{2}$ $169$ $3$ $16$ $( 1, 5, 2)( 3,10, 8)( 4, 6,11)( 9,12,13)(14,24,23)(15,20,26)(17,25,19)(18,21,22)$
3C $3^{8},1^{2}$ $338$ $3$ $16$ $( 1, 3, 8)( 2,12,11)( 5,13, 7)( 6, 9,10)(14,26,23)(15,16,19)(17,22,24)(18,25,20)$
6A1 $6^{4},2$ $507$ $6$ $21$ $( 1,23, 2,24, 5,14)( 3,25, 8,17,10,19)( 4,26,11,20, 6,15)( 7,16)( 9,18,13,22,12,21)$
6A-1 $6^{4},2$ $507$ $6$ $21$ $( 1,14, 5,24, 2,23)( 3,19,10,17, 8,25)( 4,15, 6,20,11,26)( 7,16)( 9,21,12,22,13,18)$
13A1 $13,1^{13}$ $6$ $13$ $12$ $(14,18,22,26,17,21,25,16,20,24,15,19,23)$
13A-1 $13,1^{13}$ $6$ $13$ $12$ $(14,23,19,15,24,20,16,25,21,17,26,22,18)$
13A2 $13,1^{13}$ $6$ $13$ $12$ $(14,22,17,25,20,15,23,18,26,21,16,24,19)$
13A-2 $13,1^{13}$ $6$ $13$ $12$ $(14,19,24,16,21,26,18,23,15,20,25,17,22)$
13B1 $13^{2}$ $9$ $13$ $24$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13)(14,23,19,15,24,20,16,25,21,17,26,22,18)$
13B-1 $13^{2}$ $9$ $13$ $24$ $( 1,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(14,18,22,26,17,21,25,16,20,24,15,19,23)$
13B2 $13^{2}$ $9$ $13$ $24$ $( 1, 3, 5, 7, 9,11,13, 2, 4, 6, 8,10,12)(14,19,24,16,21,26,18,23,15,20,25,17,22)$
13B-2 $13^{2}$ $9$ $13$ $24$ $( 1,12,10, 8, 6, 4, 2,13,11, 9, 7, 5, 3)(14,22,17,25,20,15,23,18,26,21,16,24,19)$
13C1 $13^{2}$ $18$ $13$ $24$ $( 1,11, 8, 5, 2,12, 9, 6, 3,13,10, 7, 4)(14,15,16,17,18,19,20,21,22,23,24,25,26)$
13C2 $13^{2}$ $18$ $13$ $24$ $( 1, 6,11, 3, 8,13, 5,10, 2, 7,12, 4, 9)(14,21,15,22,16,23,17,24,18,25,19,26,20)$
13D1 $13^{2}$ $18$ $13$ $24$ $( 1, 7,13, 6,12, 5,11, 4,10, 3, 9, 2, 8)(14,15,16,17,18,19,20,21,22,23,24,25,26)$
13D-1 $13^{2}$ $18$ $13$ $24$ $( 1, 8, 2, 9, 3,10, 4,11, 5,12, 6,13, 7)(14,18,22,26,17,21,25,16,20,24,15,19,23)$
13D2 $13^{2}$ $18$ $13$ $24$ $( 1,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(14,16,18,20,22,24,26,15,17,19,21,23,25)$
13D-2 $13^{2}$ $18$ $13$ $24$ $( 1,12,10, 8, 6, 4, 2,13,11, 9, 7, 5, 3)(14,17,20,23,26,16,19,22,25,15,18,21,24)$
26A1 $26$ $117$ $26$ $25$ $( 1,24, 2,20, 3,16, 4,25, 5,21, 6,17, 7,26, 8,22, 9,18,10,14,11,23,12,19,13,15)$
26A-1 $26$ $117$ $26$ $25$ $( 1,15,13,19,12,23,11,14,10,18, 9,22, 8,26, 7,17, 6,21, 5,25, 4,16, 3,20, 2,24)$
26A5 $26$ $117$ $26$ $25$ $( 1,16, 6,22,11,15, 3,21, 8,14,13,20, 5,26,10,19, 2,25, 7,18,12,24, 4,17, 9,23)$
26A-5 $26$ $117$ $26$ $25$ $( 1,23, 9,17, 4,24,12,18, 7,25, 2,19,10,26, 5,20,13,14, 8,21, 3,15,11,22, 6,16)$
39A1 $13,3^{4},1$ $78$ $39$ $20$ $( 1, 4, 5)( 2,13, 8)( 3, 9,11)( 6,10, 7)(14,24,21,18,15,25,22,19,16,26,23,20,17)$
39A-1 $13,3^{4},1$ $78$ $39$ $20$ $( 1, 5, 4)( 2, 8,13)( 3,11, 9)( 6, 7,10)(14,17,20,23,26,16,19,22,25,15,18,21,24)$
39A2 $13,3^{4},1$ $78$ $39$ $20$ $( 1, 5, 4)( 2, 8,13)( 3,11, 9)( 6, 7,10)(14,21,15,22,16,23,17,24,18,25,19,26,20)$
39A-2 $13,3^{4},1$ $78$ $39$ $20$ $( 1, 4, 5)( 2,13, 8)( 3, 9,11)( 6,10, 7)(14,20,26,19,25,18,24,17,23,16,22,15,21)$
39A4 $13,3^{4},1$ $78$ $39$ $20$ $( 1, 4, 5)( 2,13, 8)( 3, 9,11)( 6,10, 7)(14,15,16,17,18,19,20,21,22,23,24,25,26)$
39A-4 $13,3^{4},1$ $78$ $39$ $20$ $( 1, 5, 4)( 2, 8,13)( 3,11, 9)( 6, 7,10)(14,26,25,24,23,22,21,20,19,18,17,16,15)$
39A8 $13,3^{4},1$ $78$ $39$ $20$ $( 1, 5, 4)( 2, 8,13)( 3,11, 9)( 6, 7,10)(14,16,18,20,22,24,26,15,17,19,21,23,25)$
39A-8 $13,3^{4},1$ $78$ $39$ $20$ $( 1, 4, 5)( 2,13, 8)( 3, 9,11)( 6,10, 7)(14,25,23,21,19,17,15,26,24,22,20,18,16)$

Malle's constant $a(G)$:     $1/8$

Copy content magma:ConjugacyClasses(G);
 

Character table

35 x 35 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed